EP2299613B1 - A method for clock synchronization in a communication network - Google Patents

Abstract

The invention refers to a method for clock synchronization in a communication network comprising a plurality of nodes (MA, SL1, SL2), wherein the nodes (MA, SL1, SL2) comprise a master node (MA) and slave nodes (SL1, SL2), wherein the master node (MA) has a master clock generating master counter values and wherein each slave node (SL1, SL2) has a slave clock generating slave counter values, the clock synchronization being performed in successive synchronization cycles. In one synchronization cycle, synchronization messages (SY(k)) are transmitted in a sequence of synchronization messages (SY(k)) from the master node (MA) to a slave node (SL1) and from one slave node (SL1) to another slave node (SL2), wherein a synchronization message (SY(k)) sent from one node (MA, SL1, SL2) to another node (MA, SL1, SL2) includes information based on one or more measured counter values (TS(M(k)), TS(S 1 in (k)), TS(S 1 out (k))) of the node (MA, SL1, SL2) sending the synchronization message (SY(k)). For each slave node (SL1, SL2) the master counter value (x 1 in (k), x 2 in (k)) for a measured slave counter value (TS(S 1 in (k)), TS(S 2 in (k)) is estimated by using an estimation method applied to a probabilistic model for state variables of the nodes (MA, SL1, SL2), wherein each slave clock is synchronized based on the estimated master counter value (x 1 in (k), x 2 in (k)). A sum-product algorithm is used as the estimation method applied to the probabilistic model. The method of the invention is preferably performed in communication networks in the industrial automation domain. It enables very good clock synchronization with higher precision than standard synchronization methods.

Description

• The invention refers to a method for clock synchronization in a communication network as well as to a corresponding communication network.
• In many technical fields, communication networks are used in order to control automatically executed processes. Particularly in industrial automation networks, it is very important that the automatic processes are exactly synchronized with each other. To do so, the nodes in the communication network comprise corresponding clocks which are synchronized with a master clock of a so-called master node. The nodes having the clocks being synchronized with the master clock are usually designated as slave nodes.
• A known protocol for performing clock synchronization is the precision time protocol specified in the standard IEEE 1588. The synchronization of this protocol is based on exchanging appropriate time information by the use of synchronization messages where the time information is generated by time stamping according to the local clocks in the corresponding nodes. The synchronization messages are sequentially transmitted from one node to another node, i.e. starting from the master node to the slave node and from the slave node to the next slave node and so on until each slave node in the network has received a synchronization message. The first synchronization message sent by the master node contains a time stamp of a master counter value based on the master clock when the message is transmitted. The following slave nodes process this information and send it further. Based on the processing of a slave, all the estimated delays between the transmission time of the previous node and its own current transmission is added to the content of the synchronization message received by the slave node. Based on the information in the synchronization messages, each slave node can synchronize the slave counter values of its slave clock with the master counter values as indicated in the received synchronization messages.
• One problem in prior art synchronization methods originates from the uncertainties introduced in the time measurements, e.g. due to jitters in the time stamping, quantization errors as well as random frequency drifts of the clock frequencies. To reduce those uncertainties, it is suggested to average the results from several observations in one node. However, this is not the optimal solution because there still remains a considerable measurement error which is not acceptable for certain applications, particularly in the industrial automation domain.
• The document CHONGNING NA ET AL: "Probabilistic model for clock synchronization of cascaded network elements", IMTC'09, discloses a probabilistic model for clock synchronization using a probabilistic space model and an estimation method based on Kalman-filter to estimate a master counter value.
• Therefore, it is an object of the invention to enable clock synchronization in a communication network with higher precision than in prior art implementations.
• This object is solved by the independent claims. Preferred embodiments of the invention are defined in the dependent claims.
• The method of the invention is performed in a communication network comprising a plurality of nodes, wherein the nodes comprise a master node and slave nodes, wherein the master node has a master clock generating master counter values based on a master counter with a master frequency and wherein each slave node has a slave clock generating slave counter values based on a slave counter with the corresponding slave frequency.
• According to the method of the invention, clock synchronization is performed in successive synchronization cycles. In one synchronization cycle, synchronization messages are transmitted in a sequence of synchronization messages from the master node to a slave node and from one slave node to another slave node, wherein a synchronization message sent from one node to another node includes information based on one or more measured counter values (i.e. master or slave counter values) of the node sending the synchronization message.
• In order to obtain a precise estimation of the master counter value in each slave node, the master counter value for a measured slave counter value is estimated based on measured counter values by using a state estimation method applied to a probabilistic model for state variables of the nodes. After having estimated the master counter value, this value is used for synchronizing the corresponding slave clock.
• The invention is based on the finding that an appropriate probabilistic representation for modelling the behaviours of the nodes can be defined, thus enabling a good estimation of the master counter value used for synchronization. The inventors performed simulations and could show that the synchronization error is very low when the method of the invention is used for clock synchronization.
• In a preferred embodiment of the invention, a synchronization message sent from the master node includes the measured master counter value when the synchronization message is sent from the master node. The content of the other synchronization messages sent from one slave node to another slave node depend on the type of estimation method used according to the invention. In one embodiment of the invention, a synchronization message sent from a slave node includes an approximate master counter value when the synchronization message is sent from the slave node, said approximate master counter value being derived from measured slave counter values of the slave node and from information received in synchronization messages. This form of synchronization message is preferably used in case that the Kalman filter mentioned below is used as the estimation method.
• In another embodiment of the invention, a synchronization message sent from a slave node includes the measured bridge delay between receiving the previous synchronization message and sending the synchronization message as well as the estimated master counter value for this slave node. This kind of synchronization message is preferably used when the estimation method is based on the sum-product algorithm described below.
• In a preferred embodiment of the invention, the probabilistic model incorporates measurement errors with respect to measured counter values as well as variations of the frequency ratio between the master clock and each slave clock.
• In a particularly preferred embodiment of the invention, the probabilistic model includes a state transition model as well as an observation model. The state transmission model describes the dependency of state variables of slave nodes in a synchronization cycle on state variables of slave nodes in the previous synchronization cycle. The observation model describes the dependency of known variables of master and slave nodes on state variables of slave nodes, particularly at the time of receiving a synchronization message. Here and in the following, the term "known variables" refer to variables which are measured or determined with a certain accuracy in a separate process. By using these models, the state variables of the slave nodes are estimated by the estimation method.
• In a preferred embodiment of the aforementioned probabilistic model including a state transition model and an observation model, the state variables comprise the master counter value and the frequency ratio between the master frequency and a slave frequency when the synchronization message is received in a slave clock. Preferably, the state transition model is defined as follows: $$R_n\left(k\right)=R_n\left(k-1\right)+{\omega }_n\left(k\right)$$

  

  $$x_n^{\mathrm{in}}\left(k\right)=x_n^{\mathrm{in}}\left(k-1\right)+a_n\left(k\right)\cdot R_n\left(k-1\right),$$

  

  where n refers to the nth slave node receiving a synchronization message in the sequence of synchronization messages;
  where k refers to the kth synchronization cycle;
  where $x_n^{\mathrm{in}}\left(k\right)$

  

  refers to the master counter value at the time of receiving the synchronization message in the nth slave node in the kth synchronization cycle;
  where R_n (k) is the frequency ratio between the master frequency and the slave frequency of the nth slave node in the kth synchronization cycle;
  where ω _n (k) refers to a random variable modelling frequency variations, where the random variable is preferably modelled as a Gaussian distribution;
  where a_n (k) is the time interval measured in slave counter values between receiving the synchronization message in the nth slave node in the kth synchronization cycle and receiving the corresponding synchronization message in the nth slave node in the (k-1)th synchronization cycle.
• In another preferred embodiment of the invention, the known state variables used in the observation model comprise a separately estimated line delay (i.e. a line delay being determined with a certain accuracy in a separate line estimation process) in slave counter values between receiving successive synchronization messages in a slave node, a measured bridge delay in slave counter values between receiving a synchronization message and sending the next synchronization message in a slave node and the measured master counter value when the master node sends a synchronization message. Preferably, the observation model is defined as follows: $$\mathit{TS}\left(M\left(k\right)\right)=x_1^{\mathrm{in}}\left(k\right)-c_1\left(k\right)\cdot R_1\left(k\right)+{\nu }_1\left(k\right)$$

  

  $$x_{n-1}^{\mathrm{in}}\left(k\right)+b_{n-1}\left(k\right)\cdot R_{n-1}\left(k\right)=x_n^{\mathrm{in}}\left(k\right)-c_n\left(k\right)\cdot R_n\left(k\right)+{\nu }_n\left(k\right)\mathrm{for}n>1$$

  

  
  where n refers to the nth slave node receiving a synchronization message in the sequence of synchronization messages;
  where k refers to the kth synchronization cycle;
  where TS(M(k)) refers to the measured master counter value when the master node sends the synchronization message in the kth synchronization cycle;
  where $x_n^{\mathrm{in}}\left(k\right)$

  

  refers to the master counter value at the time of receiving the synchronization message in the nth slave node in the kth synchronization cycle;
  where R_n (k) is the frequency ratio between the master frequency and the slave frequency of the nth slave node in the kth synchronization cycle;
  where c_n (k) refers to the separately estimated line delay in slave counter values between receiving successive synchronization messages in the nth slave node in the kth synchronization cycle;
  where b_n (k) refers to the measured bridge delay in slave counter values between receiving the synchronization message and sending the next synchronization message in the nth slave node in the kth synchronization cycle;
  where v_n (k) refers to a random variable modelling measurement errors in the nth slave node, said random variable preferably being modelled by a Gaussian distribution, said distribution preferably being generated from test sample data.
• The above described formulas for the state transition model and the observation model have been obtained by a complex mathematic derivation explained in detail in the detailed description.
• In a preferred embodiment of the invention, the estimation method for estimating the master counter value is based on the Kalman filter which is well known from the prior art. In the detailed description of the invention, the application of the Kalman filter to the above defined state transition model and observation model is explained.
• In another embodiment of the invention, particularly in the embodiment using the Kalman filter, in each synchronization cycle the known state variables of each node are transmitted to a common fusion center in which based on the state transition model and the observation model the state variables are estimated by the estimation method, said estimated state variables being transmitted to the corresponding nodes in order to synchronize the slave clocks.
• In another embodiment of the invention, particularly in the embodiment using the sum-product algorithm described below, the state variables of a slave node are estimated in the slave node.
• As mentioned before, the estimation method in one embodiment of the invention is based on the so-called sum-product algorithm which is per se known from the prior art. This sum-product algorithm is applied to a factor graph describing the relationships between known variables and the state variables based on message flows. In a particularly preferred embodiment, the factor graph has constraints with respect to the message flows such that messages are not passed from a node receiving a synchronization message to a node from which the synchronization message is received and such that messages are not passed back in time. This enables an implementation of the method according to the invention where state variables are estimated directly in each node.
• The clock synchronization used according to the invention is preferably based on the above mentioned standard IEEE 1588 describing the exchange of synchronization messages. In another embodiment of the invention, the nodes in the communication network communicate with each other based on the well-known PROFINET standard. Furthermore, the method according to the invention is preferably used in a communication network in an industrial automation system.
• Besides the above method, the invention also refers to a communication network comprising a plurality of nodes where the nodes are adapted to perform a clock synchronization based on the synchronization method of the invention.
• Embodiments of the invention will now be described in detail with respect to the accompanying drawings, wherein:

Fig. 1

shows a schematic view on several network nodes exchanging synchronization messages according to an embodiment of the invention;

Fig. 2

shows a diagram indicating the propagation of synchronization messages in an embodiment according to the invention;

Fig. 3

shows a diagram with respect to an embodiment of the invention using a fusion center for estimating state variables;

Fig. 4 and 5A to 5B

show factor graphs used by the sum- product algorithm according to an embodiment of the invention; and

Fig. 6

shows a diagram indicating the information carried by the synchronization messages according to an embodiment of the invention.

• Fig. 1 shows a chain of nodes in a communication network performing the method of the invention. The communication network comprises a master node MA as well as several slave nodes, where in Fig. 1 slave nodes SL1 and SL2 are shown. The master node has a master clock and each slave node has a slave clock. In the communication network according to Fig. 1, an appropriate clock synchronization protocol, e.g. the precision time protocol according to the standard IEEE 1588, is used for synchronizing the clock of each slave node with the clock of the master node MA. To do so, synchronization messages SY(k) are forwarded from one node to another, i.e. from the master node MA to the slave node SL1, from the slave node SL1 to the slave node SL2 and so on until the last slave node SLN in the chain of slave nodes is reached. The transmission of synchronization messages is repeated in successive synchronization cycles, where k indicates the current number of the synchronization cycle. In order to perform synchronization, the messages SY(k) include time stamps indicating the master counter value of the master clock at the time of sending the synchronization message. I.e., the master counter value in a message is updated in each slave node by adding to the time of the master clock of a received synchronization message the time interval from receiving the synchronization message until sending the next synchronization message. The time interval being added is subject to measurement errors which is evident from the diagram of Fig. 2.
• In Fig. 2, the three vertical lines refer to the time being measured in the master node MA, the slave node SL1 and the slave node SL2, respectively. The time axis according to those vertical lines extends from above to below, i.e. future events are at lower positions along the vertical lines. The clock of the master node MA works at a master frequency and the clocks of the slave nodes SL1 and SL2 work at respective slave frequencies which can be different from each other and from the master frequency. The time for each node is measured in the corresponding clock of the node, i.e. with the corresponding frequency of the clock of the respective node.
• In the diagram of Fig. 2, the function TS refers to the corresponding time stamp of a counter and indicates the measured value of the counter. Due to stamping jitters and frequency drifts, the time stamp differs from the true counter value. Fig. 2 shows the transmission of the synchronization message SY(k) which is sent by the master node MA at the true master counter value M(k) corresponding to the measured master counter value TS(M(k)). This message is received in slave node SL1 at the slave counter value $S_{}^{}$ ${}_1^{\mathit{in}}\left(k\right)$

  

  which corresponds to the measured slave counter value $\mathit{TS}\left(S_{}^{}\right)$ $\left({}_1^{\mathit{in}}\left(k\right)\right)\mathrm{.}$

  

  The time between sending the message from the master node MA and receiving the message in the slave node SL1 is called line delay and is estimated by a corresponding estimation process. Particularly, in the standard IEEE 1588 such an estimation process is described and may also be used in the method according to the invention. Upon receipt of the synchronization message and an additional bridge delay, a new synchronization message is sent from slave node SL1 to slave node SL2. The new message is sent at the slave counter value $S_{}^{}$ ${}_1^{\mathit{out}}\left(k\right)$

  

  corresponding to a measured slave counter value $\mathit{TS}\left(S_{}^{}\right)$ $\left({}_1^{\mathit{out}}\left(k\right)\right)\mathrm{.}$

  
• The synchronization message sent from the slave node SL1 to the slave node SL2 is received in the slave node SL2 at the true slave counter value $S_{}^{}$ ${}_2^{\mathit{in}}\left(k\right)$

  

  corresponding to the measured slave counter value $\mathit{TS}\left(S_{}^{}\right)$ $\left({}_2^{\mathit{in}}\left(k\right)\right)\mathrm{.}$

  

  Thereafter, the method repeats and a new synchronization message SY(k) will be sent to the next slave node and so on until all N slave nodes in the network have received a synchronization message. As is evident from Fig. 2, the measured slave counter values (i.e. the time stamps TS of the slave counter values) have corresponding master counter values indicated by time variables beginning with the letter "x" on the first vertical line. Analogously, the true slave counter values have corresponding master counter values indicated by time variables beginning with the letter "M" on the first vertical line.
• One problem arising in synchronization are the uncertainties introduced due to measurement errors, like stamping jitter, quantization errors and random frequency drifts between the clocks of the slave nodes and the master clock. Due to these errors, the synchronization of the clocks is not exact which may result in problems in some applications where combined processes needing an exact matching of the clocks are performed by the nodes in the network. Particularly, in applications referring to industrial automation processes, an exact synchronization of the clocks between the nodes is very important.
• According to the invention, a probabilistic model representing the relationships between state variables of the different nodes is introduced. This model provides a good estimation of the counter values in the respective nodes such that a better synchronization of the slave clocks with respect to the master clock can be established. In the following, an appropriate probabilistic model will be derived. Thereafter, this probabilistic model will be used by two different estimation methods for estimating the states of the nodes. One estimation method which refers to a first embodiment of the invention is based on the Kalman filter and the other estimation method being a second embodiment of the invention is based on the sum-product algorithm.
• At first, the variables appearing in Fig. 2 for the general case of a slave node SLn (n = 1, ..., N) will be explained. In the terminology used in the following, a synchronization message will also be called Sync message and a slave node SLn will also be designated as slave n. Furthermore, the term "master" is used for indicating the master node MA.
• TS (x): time-stamp of counter with value x.
  ξ(x): stamping error, i.e., $$\mathit{TS}\left(x\right)=x+\xi \left(x\right)$$

  

  
  R _n (k) : the k^th RCF, i.e. frequency ratio of master frequency and slave n's frequency.
  M(k) : master counter value when the k^th Sync message is transmitted by the master (which is stamped as, i.e. believed to be TS(M(k))).
  $S_n^{\mathit{in}}\left(k\right):$

  

  counter value of slave n when the k^th Sync message arrives at slave n (which is stamped as, i.e. believed to be $\mathit{TS}\left(S_n^{\mathit{in}}\left(k\right)\right))\mathrm{.}$

  

  
  $M_n^{\mathit{in}}\left(k\right):$

  

  master counter value at this time, i.e. for slave n's counter value $S_n^{\mathit{in}}\left(k\right)\mathrm{.}$

  

  
  $x_n^{\mathit{in}}\left(k\right):$

  

  master counter at the time when slave n's counter really is $\mathit{TS}\left(S_n^{\mathit{in}}\left(k\right)\right)\mathrm{.}$

  

  
  $S_n^{\mathit{out}}\left(k\right):$

  

  counter value of slave n when the k^th Sync message is forwarded by slave n (which is stamped as, i.e. believed to be $\mathit{TS}\left(S_n^{\mathit{out}}\left(k\right)\right))\mathrm{.}$

  

  
  $M_n^{\mathit{out}}\left(k\right):$

  

  master counter value at this time, i.e. for slave n's counter value $S_n^{\mathit{out}}\left(k\right)\mathrm{.}$

  

  
  $x_n^{\mathit{out}}\left(k\right):$

  

  master counter at the time when slave n's counter really is $\mathit{TS}\left(S_n^{\mathit{out}}\left(k\right)\right)\mathrm{.}$

  

  
  M(LD_n ): the line delay between slave n-1 and slave n (for n = 1 between master and slave 1), measured by the master clock.
  S_n (LD_n ): the line delay between slave n-1 and slave n (for n = 1 between master and slave 1), measured by slave n's clock.
  Ŝ_n (LD_n ) : estimate of the line delay between slave n-1 and slave n (for n = 1 between master and slave 1), measured by slave n's clock.
  $\eta \left({\hat{S}}_n\left({\mathit{LD}}_n\right)\right):$

  

  estimation error of Ŝ_n (LD_n ), i.e. $${\hat{S}}_n\left({\mathit{LD}}_n\right)=S_n\left({\mathit{LD}}_n\right)+\eta \left({\hat{S}}_n\left({\mathit{LD}}_n\right)\right)\mathrm{.}$$

  
• In the following, an appropriate state transition model and observation model will be derived. As mentioned above, those models are used by two embodiments of the invention described later on for estimating the master counter in each slave.
• The state transition model of RCF is given by: $$R_n\left(k\right)=R_n\left(k-1\right)+{\omega }_n\left(k\right)$$

  

  where ω _n (k) is the process noise of RCF. If the frequency drifts, the performance is sensitive to the choice of ω _n (k). However, the following derivation is focussed on reducing the influence of stamping noise in the case of constant frequency.
• The state transition of $x_k^{\mathit{in}}\left(k\right)$

  

  is given by: $$x_n^{\mathrm{in}}\left(k\right)=x_n^{\mathrm{in}}\left(k-1\right)+\left(\mathit{TS}\left(S_n^{\mathrm{in}}\left(k\right)\right)-\mathit{TS}\left(S_n^{\mathrm{in}}\left(k-1\right)\right)\right)\cdot R_n\left(k-1\right)$$

  
• Using a_n (k) to denote $\mathit{TS}\left(S_n^{\mathit{in}}\left(k\right)\right)-\mathit{TS}\left(S_n^{\mathit{in}}\left(k-1\right)\right),$

  

  then (3) can be written as: $$\begin{array}{|c|}\hline x_n^{\mathrm{in}}\left(k\right)=x_n^{\mathrm{in}}\left(k-1\right)+a_n\left(k\right)\cdot R_n\left(k-1\right)\\ \hline\end{array}$$

  
• $x_k^{\mathit{in}}\left(k\right)$

  

  and $x_k^{\mathit{out}}\left(k\right)$

  

  are related by: $$x_n^{\mathrm{out}}\left(k\right)=x_n^{\mathrm{in}}\left(k\right)+\left(\mathit{TS}\left(S_n^{\mathrm{out}}\left(k\right)\right)-\mathit{TS}\left(S_n^{\mathrm{in}}\left(k\right)\right)\right)\cdot R_n\left(k\right)$$

  
• Using b_n (k) to denote $\mathit{TS}\left(S_n^{\mathit{out}}\left(k\right)\right)-\mathit{TS}\left(S_n^{\mathit{in}}\left(k\right)\right)$

  

  and rewrite (5) : $$\begin{array}{|c|}\hline x_n^{\mathrm{out}}\left(k\right)=x_n^{\mathrm{in}}\left(k\right)+b_n\left(k\right)\cdot R_n\left(k\right)\\ \hline\end{array}$$

  
• M (k) and ${\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">M</figure-callout>}}_1^{\mathit{in}}\left(k\right)$

  

  are related by the line delay between master and slave 1: $$\begin{array}{ll}M\left(k\right)& ={\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">M</figure-callout>}}_1^{\mathrm{in}}\left(k\right)-\mathrm{<figure-callout\; id="1"\; label="M\; LD"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">M</figure-callout>}\left({\mathit{LD}}_{}\right)\left({\mathit{}}_1\right)\\ & ={\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">M</figure-callout>}}_1^{\mathrm{in}}\left(k\right)-{\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">S</figure-callout>}}_1\left({\mathit{<figure-callout\; id="1"\; label="LD"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">LD</figure-callout>}}_1\right)\cdot R_1\left(k\right)\\ & ={\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">M</figure-callout>}}_1^{\mathrm{in}}\left(k\right)-\left({\hat{S}}_1\left({\mathit{LD}}_1\right)-\eta \left({\hat{S}}_1\left({\mathit{<figure-callout\; id="1"\; label="LD"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">LD</figure-callout>}}_1\right)\right)\right)\cdot R_1\left(k\right)\end{array}$$

  
• For n>1, $M_{n-1}^{\mathit{out}}\left(k\right)$

  

  and ${\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">M</figure-callout>}}_1^{\mathit{in}}\left(k\right)$

  

  are related by: , $$\begin{array}{ll}{M}_{n-1}^{\mathrm{out}}\left(k\right)& =M_n^{\mathrm{in}}\left(k\right)-M\left({\mathit{LD}}_n\right)=M_n^{\mathrm{in}}\left(k\right)-S_n\left({\mathit{LD}}_n\right)\cdot R_n\left(k\right)\\ & =M_n^{\mathrm{in}}\left(k\right)-\left({\hat{S}}_n\left({\mathit{LD}}_n\right)-\eta \left({\hat{S}}_n\left({\mathit{LD}}_n\right)\right)\right)\cdot R_n\left(k\right)\end{array}$$

  
• For all TT ∈ {in, out} the following relation holds by definition of R_n(k): $$x_n^{\mathrm{TT}}\left(k\right)-M_n^{\mathrm{TT}}\left(k\right)=R_n\left(k\right)\cdot \left(\mathit{TS}\left(S_n^{\mathrm{TT}}\left(k\right)\right)-S_n^{\mathrm{TT}}\left(k\right)\right)=R_n\left(k\right)\cdot \xi \left(S_n^{\mathrm{TT}}\left(k\right)\right)$$

  
• For deriving this relation, equation (1) with argument $S_n^{\mathrm{TT}}\left(k\right)$

  

  is inserted in the second equality of (9).
• Applying equation (1) with argument M (k) on the left hand side of equation (7) and applying equation (9) on the 1st term of the right hand side of equation (7), this equation can be rewritten as: $$\begin{array}{ll}& \mathit{TS}\left(M\left(k\right)\right)-\xi \left(M\left(k\right)\right)=\left(x_1^{\mathrm{in}}\left(k\right)-R_1\left(k\right)\cdot \mathrm{<figure-callout\; id="1"\; label="\xi \; S"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">\xi </figure-callout>}\left(S_{}^{}\right)\left({}_1^{\mathrm{in}}\left(k\right)\right)\right)-\left({\hat{S}}_1\left({\mathit{LD}}_1\right)-\eta \left({\hat{S}}_1\left({\mathit{<figure-callout\; id="1"\; label="LD"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">LD</figure-callout>}}_1\right)\right)\right)\cdot R_1\left(k\right)\\ \Rightarrow & \mathit{TS}\left(M\left(k\right)\right)=x_1^{\mathrm{in}}\left(k\right)-{\hat{S}}_1\left({\mathit{<figure-callout\; id="1"\; label="LD"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">LD</figure-callout>}}_1\right)\cdot R_1\left(k\right)+\eta \left({\hat{S}}_1\left({\mathit{<figure-callout\; id="1"\; label="LD"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">LD</figure-callout>}}_1\right)\right)\cdot R_1\left(k\right)+\xi \left(M\left(k\right)\right)-R_1\left(k\right)\cdot \mathrm{<figure-callout\; id="1"\; label="\xi \; S"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">\xi </figure-callout>}\left(S_{}^{}\right)\left({}_1^{\mathrm{in}}\left(k\right)\right)\end{array}$$

  
• Using equation (9) on the left hand side and on the 1st term of the right hand side of equation (8), this equation can be rewritten as: $$\begin{array}{ll}& x_{n-1}^{\mathrm{out}}\left(k\right)-R_{n-1}\left(k\right)\cdot \xi \left(S_{n-1}^{\mathrm{out}}\left(k\right)\right)=\left(x_n^{\mathrm{in}}\left(k\right)-R_n\left(k\right)\cdot \xi \left(S_n^{\mathrm{in}}\left(k\right)\right)\right)-{\hat{S}}_n\left({\mathit{LD}}_n\right)\cdot R_n\left(k\right)+\eta \left({\hat{S}}_n\left({\mathit{LD}}_n\right)\right)\cdot R_n\left(k\right)\\ \Rightarrow & x_{n-1}^{\mathrm{out}}\left(k\right)=x_n^{\mathrm{in}}\left(k\right)-{\hat{S}}_n\left({\mathit{LD}}_n\right)\cdot R_n\left(k\right)+\eta \left({\hat{S}}_n\left({\mathit{LD}}_n\right)\right)\cdot R_n\left(k\right)+R_{n-1}\left(k\right)\cdot \xi \left(S_{n-1}^{\mathrm{out}}\left(k\right)\right)-R_n\left(k\right)\cdot \xi \left(S_n^{\mathrm{in}}\left(k\right)\right)\end{array}$$

  
• Using equation (6), equation (11) can be rewritten as: $$\begin{array}{ll}{x}_{n-1}^{\mathrm{in}}\left(k\right)+b_{n-1}\left(k\right)\cdot R_{n-1}\left(k\right)& =x_n^{\mathrm{in}}\left(k\right)-{\hat{S}}_n\left({\mathit{LD}}_n\right)\cdot R_n\left(k\right)+\\ & +\eta \left({\hat{S}}_n\left({\mathit{LD}}_n\right)\right)\cdot R_n\left(k\right)+R_{n-1}\left(k\right)\cdot \xi \left(S_{n-1}^{\mathrm{out}}\left(k\right)\right)-R_n\left(k\right)\cdot \xi \left(S_n^{\mathrm{in}}\left(k\right)\right)\end{array}$$

  
• By defining: $$v_n\left(k\right)=\{\begin{array}{cc}\eta \left({\hat{S}}_1\left({\mathit{<figure-callout\; id="1"\; label="LD"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">LD</figure-callout>}}_1\right)\right)\cdot R_1\left(k\right)+\xi \left(M\left(k\right)\right)-R_1\left(k\right)\cdot \mathrm{<figure-callout\; id="1"\; label="\xi \; S"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">\xi </figure-callout>}\left(S_{}^{}\right)\left({}_1^{\mathrm{in}}\left(k\right)\right)& n=1\\ \eta \left({\hat{S}}_n\left({\mathit{LD}}_n\right)\right)\cdot R_n\left(k\right)+R_{n-1}\left(k\right)\cdot \xi \left(S_{n-1}^{\mathrm{out}}\left(k\right)\right)-R_n\left(k\right)\cdot \xi \left(S_n^{\mathrm{in}}\left(k\right)\right)& n>1\end{array}$$

  

  and c_n (k) = Ŝ(LD_n )
• equations (10) and (12) can be rewritten as follows: $$\begin{array}{|c|}\hline \mathit{TS}\left(M\left(k\right)\right)=x_1^{\mathrm{in}}\left(k\right)-c_1\left(k\right)\cdot R_1\left(k\right)+{\nu }_1\left(k\right)\\ \hline\end{array}$$

  

  $$\begin{array}{|c|}\hline x_{n-1}^{\mathrm{in}}\left(k\right)+b_{n-1}\left(k\right)\cdot R_{n-1}\left(k\right)=x_n^{\mathrm{in}}\left(k\right)-c_n\left(k\right)\cdot R_n\left(k\right)+{\nu }_n\left(k\right)\left(\mathrm{for}n>1\right)\\ \hline\end{array}$$

  
• The most important formulas that have been derived are as follows: $$R_n\left(k\right)=R_n\left(k-1\right)+{\omega }_n\left(k\right)$$

  

  $$x_n^{\mathrm{in}}\left(k\right)=x_n^{\mathrm{in}}\left(k-1\right)+a_n\left(k\right)\cdot R_n\left(k-1\right)$$

  

  $$\mathit{TS}\left(M\left(k\right)\right)=x_1^{\mathrm{in}}\left(k\right)-c_1\left(k\right)\cdot R_1\left(k\right)+{\nu }_1\left(k\right)$$

  

  $$x_{n-1}^{\mathrm{in}}\left(k\right)+b_{n-1}\left(k\right)\cdot R_{n-1}\left(k\right)=x_n^{\mathrm{in}}\left(k\right)-c_n\left(k\right)\cdot R_n\left(k\right)+{\nu }_n\left(k\right)\mathrm{for}n>1$$

  
• The variables above fall into the following categories: TS(M(k)), a_n (k), (n = 1,...N), c_n (k), (n = 1,...N) and b _n-1(k), (n = 2,...N) are known variables ( i.e. variables which are measured or determined with a certain accuracy in a separate process), $x_n^{\mathrm{in}}\left(k\right)$

  

  and R_n (k) are hidden state variables which need to be estimated, and v_n (k), ω _n (k) are random variables.
• The first two formulas (2) and (4) reveal the time correlation of the state variables, so they constitute the state transition model. The other two formulae (14) and (15) reveal the space correlation of the state variables; hence they are the observation or coupling model. These four equations represent the state-space-model of the system.
• The above derived state transition model and observation model will be used in both embodiments described in the following for estimating the state variables $x_n^{\mathrm{in}}\left(k\right)$

  

  and R _n (k) (also denoted as hidden state variables) used for clock synchronization.
• The first embodiment uses a Kalman filter for estimating the variables and is based on the following derivation:
• Using (2) and (4), the following state transition model for the whole system is obtained: $$\left[\begin{array}{c}{x}_1^{\mathrm{in}}\left(k\right)\\ ⋮\\ x_N^{\mathrm{in}}\left(k\right)\\ R_1\left(k\right)\\ ⋮\\ R_N\left(k\right)\end{array}\right]=\left[\begin{array}{cccccc}1& \cdots & 0& a_1\left(k\right)& \cdots & 0\\ ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ 0& \cdots & 1& 0& \cdots & a_N\left(k\right)\\ 0& \cdots & 0& 1& \cdots & 0\\ ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ 0& \cdots & 0& 0& \cdots & 1\end{array}\right]\left[\begin{array}{c}{x}_1^{\mathrm{in}}\left(k-1\right)\\ ⋮\\ x_N^{\mathrm{in}}\left(k-1\right)\\ {\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">R</figure-callout>}}_1\left(k-1\right)\\ ⋮\\ R_N\left(k-1\right)\end{array}\right]+\left[\begin{array}{c}0\\ ⋮\\ 0\\ {\omega }_1\left(k\right)\\ ⋮\\ {\omega }_N\left(k\right)\end{array}\right]$$

  
• Using x(k) to denote the vector of hidden state variables on the left hand side of (16), using A(k) for the first matrix on the right hand side, and ω(k)=[ω _i (k)...ω _N (k)] ^T for the noise vector and defining E _N =[0 _N ,I _N ] where 0 _N is an N×N null matrix and I _N is an identity matrix of dimension N, we can write (16) as follows: $$\begin{array}{|c|}\hline \mathbf{x}\left(k\right)=\mathbf{A}\left(k\right)\cdot \mathbf{x}\left(k-1\right)+{\mathbf{E}}_N^{\mathrm{T}}\cdot \mathbf{\omega }\left(k\right)\\ \hline\end{array}$$

  

  which gives the state transition model.
• Equation (15) can be rewritten in the following form: $$\begin{array}{cc}0=-x_{n-1}^{\mathrm{in}}\left(k\right)+x_n^{\mathrm{in}}\left(k\right)-b_{n-1}\left(k\right)\cdot R_{n-1}\left(k\right)-c_n\left(k\right)\cdot R_n\left(k\right)+{\nu }_n\left(k\right)& \mathrm{for}n\ge 2\end{array}$$

  
• Combining (14) with (18) for all elements, the following equation is obtained: $$\left[\begin{array}{c}\mathit{TS}\left(M\left(k\right)\right)\\ 0\\ ⋮\\ 0\end{array}\right]=\left[\begin{array}{cccccccccc}1& 0& \cdots & 0& 0& -c_1\left(k\right)& 0& \cdots & 0& 0\\ -1& 1& \cdots & 0& 0& -b_1\left(k\right)& -c_2\left(k\right)& \cdots & 0& 0\\ ⋮& ⋮& \ddots & ⋮& ⋮& ⋮& ⋮& \ddots & ⋮& ⋮\\ 0& 0& \cdots & -1& 1& 0& 0& \cdots & -b_{N-1}\left(k\right)& -c_N\left(k\right)\end{array}\right]\left[\begin{array}{c}{x}_1^{\mathrm{in}}\left(k\right)\\ x_2^{\mathrm{in}}\left(k\right)\\ ⋮\\ x_{N-1}^{\mathrm{in}}\left(k\right)\\ x_N^{\mathrm{in}}\left(k\right)\\ R_1\left(k\right)\\ R_2\left(k\right)\\ ⋮\\ R_{N-1}\left(k\right)\\ R_N\left(k\right)\end{array}\right]+\left[\begin{array}{c}{\nu }_1\left(k\right)\\ {\nu }_2\left(k\right)\\ ⋮\\ {\nu }_N\left(k\right)\end{array}\right]$$

  
• Using C(k) to denote the matrix on the right hand side of (19), y _N for the left hand side, ν _N for the noise vector, equation (18) can be rewritten as follows: $$\begin{array}{|c|}\hline \mathbf{y}\left(k\right)=\mathbf{C}\left(k\right)\cdot \mathbf{x}\left(k\right)+\mathbf{\nu }\left(k\right)\\ \hline\end{array}$$

  

  which gives the observation model.
• Based on the above derivation two important formulas are obtained: $$\mathbf{x}\left(k\right)=\mathbf{A}\left(k\right)\cdot \mathbf{x}\left(k-1\right)+{\mathbf{E}}_N^{\mathrm{T}}\cdot \mathbf{\omega }\left(k\right)$$

  

  $$\mathbf{y}\left(k\right)=\mathbf{C}\left(k\right)\cdot \mathbf{x}\left(k\right)+\mathbf{\nu }\left(k\right)$$

  
• The vectors and matrices in these two formulas fall into the following categories:
• y ₍ k), A(k) and C(k) are known,
• x(k) contains the hidden state variables,
• ω(k) and ν(k) are random variables (with distributions about which certain assumptions can be made).
• The first formula reveals the time correlation of the state variables, so it constitutes the state transition model. The second one reveals the space correlation of the state variables; hence it is the observation or coupling model. These two equations represent the state-space-model of the system.
• The random vectors x(k) and x(k-1) are related via the probability density function p _∞(k)(ω(k)) of the noise ω(k); the distribution of x(k) is given by the probability density function p _v(k)(v(k)) of the noise v(k).
• It is assumed that all noises are Gaussian noise, i.e. ${\omega }_n\left(k\right)\sim N\left(0,{\sigma }_{{\omega }_n\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2\right)$

  

  and ${\nu }_n\left(k\right)\sim N\left(0,{\sigma }_{{\nu }_n\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2\right)$

  

  for all n ∈ {1,...,N}. The values of the noise variance are obtained from the definition of the stamping jitters of the hardware of the nodes. R(k) and Q(k) is used to denote the covariance matrix of ν(k) and ω(k), so: $$\mathbf{R}\left(k\right)=\left[\begin{array}{ccc}{\sigma }_{{\nu }_1\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & {\sigma }_{{\nu }_N\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2\end{array}\right]$$

  

  and $$\mathbf{Q}\left(k\right)=\left[\begin{array}{ccc}{\sigma }_{{\omega }_1\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & {\sigma }_{{\omega }_N\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2\end{array}\right]$$

  
• R(k) and Q(k) can also be time-independent, i.e. independent of k, as exemplified below.
• As we can see from (13), ν _n (k) is a linear combination of many stamping jitters with each jitter being a random variable generated from a distribution. Usually, the distribution function of the jitters is obtained from the description of the hardware of the nodes or from experiments. Given all the distributions, a Monte Carlo method may be used to generate many samples of jitters.
• Using (13), many samples ${\left\{\begin{array}{c}{r}_{{\mathit{\nu }}_n\left(k\right)}^{\left(l\right)}\end{array}\right\}}_{l=1}^L$

  

  of ν _n (k) can be produced based on the samples of the jitters. Then the best Gaussian distribution p _{ν n (k)}(ν _n (k)) can be determined that fits the histogram of the data ${\left\{\begin{array}{c}{r}_{{\mathit{\nu }}_n\left(k\right)}^{\left(l\right)}\end{array}\right\}}_{l=1}^L$

  

  and thus the value of σ_{ν n (k)} can be obtained.
• The state transition model of the RCF in (2) indicates that the underlying model assumes constant frequency and uses ω _n (k) to control the strength of this assumption. If the frequency changes, this will result in a model mismatch problem. It can be determined how much such a model should be trusted by adjusting the value of σ_{ω n (k)}. It balances the importance between the state transition model and the measurements. If σ_{ω n (k)} is big, then the error introduced by model mismatch is decreased by trusting the measurements. However, the noise made in the observation is tolerated. If σ_{ω n (k)} is small, the estimate of frequency is forced to be constant. By doing that, the risk of model mismatch will rise. However, if the frequency is constant, small σ_{ω n (k)} can minimize the influence of the observation noise. In practice, the choice of the value of σ_{ω n (k)} should be based on the stability of the oscillator used for the clocks of the nodes and the magnitude of the stamping jitters.
• Then $$p_{\mathbf{\omega }\left(k\right)}\left(\mathbf{\omega }\left(k\right)\right)=\frac{1}{{\left(2\pi \right)}^{N/2}{\left|\mathbf{<figure-callout\; id="1"\; label="Q\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">Q</figure-callout>}\left(k\right)\left(\right)\right|}^{1/2}}\exp \left(-\frac{1}{2}{\mathbf{\omega }}^{\mathrm{T}}\left(k\right){\mathbf{Q}}^{-1}\left(k\right)\mathbf{\omega }\left(k\right)\right)$$

  

  and since from (17): $$\mathbf{\omega }\left(k\right)={\mathbf{E}}_N\cdot \left(\mathbf{x}\left(k\right)-\mathbf{A}\left(k\right)\cdot \mathbf{x}\left(k-1\right)\right)$$

  

  leads to: $$\begin{array}{l}{p}_{\omega \left(k\right)}\left(\mathbf{x}\left(k\right)-\mathbf{A}\left(k\right)\cdot \mathbf{x}\left(k-1\right)\right)=\\ =\frac{1}{{\left(2\pi \right)}^{N/2}{\left|\mathbf{<figure-callout\; id="1"\; label="Q\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">Q</figure-callout>}\left(k\right)\left(\right)\right|}^{1/2}}\exp \left[-\frac{1}{2}{\left(\mathbf{x}\left(k\right)-\mathbf{A}\left(k\right)\cdot \mathbf{x}\left(k-1\right)\right)}^{\mathrm{T}}{\mathbf{E}}_N^{\mathrm{T}}{\mathbf{Q}}^{-1}\left(k\right){\mathbf{E}}_N\left(\mathbf{x}\left(k\right)-\mathbf{A}\left(k\right)\cdot \mathbf{x}\left(k-1\right)\right)\right]\end{array}$$

  
• Likewise: $$p_{\mathbf{\nu }\left(k\right)}\left(\mathbf{\nu }\left(k\right)\right)=\frac{1}{{\left(2\pi \right)}^{N/2}{\left|\mathbf{<figure-callout\; id="1"\; label="R\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">R</figure-callout>}\left(k\right)\left(\right)\right|}^{1/2}}\exp \left(-\frac{1}{2}\mathbf{\nu }{\left(k\right)}^{\mathrm{T}}{\mathbf{R}}^{-1}\left(k\right)\mathbf{\nu }\left(k\right)\right)$$

  

  and from (20): $$\mathbf{\nu }\left(k\right)=\mathbf{y}\left(k\right)-\mathbf{C}\left(k\right)\cdot \mathbf{x}\left(k\right)$$

  

  leads to: $$\begin{array}{l}{p}_{\upsilon \left(k\right)}\left(\mathbf{y}\left(k\right)-\mathbf{C}\left(k\right)\cdot \mathbf{x}\left(k\right)\right)=\\ =\frac{1}{{\left(2\pi \right)}^{N/2}{\left|\mathbf{<figure-callout\; id="1"\; label="R\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">R</figure-callout>}\left(k\right)\left(\right)\right|}^{1/2}}\exp \left[-\frac{1}{2}{\left(\mathbf{y}\left(k\right)-\mathbf{C}\left(k\right)\cdot \mathbf{x}\left(k\right)\right)}^{\mathrm{T}}{\mathbf{R}}^{-1}\left(k\right)\left(\mathbf{y}\left(k\right)-\mathbf{C}\left(k\right)\cdot \mathbf{x}\left(k\right)\right)\right]\end{array}$$

  
• Based on the above derivation, the well-known Kalman filter is used in order to estimate the hidden variables, i.e. the variables in x(k), in equation (17) and (20). The calculation is summarized as follows:
• Prediction step:
  • predicted state: x̂(k,k-1)=A(k)·x̂(k-1, k-1)
  • predicted estimate covariance: $$\mathbf{P}\left(k,k-1\right)=\mathbf{A}\left(k\right)\cdot \mathbf{P}\left(k-1,k-1\right)\cdot {\mathbf{A}}^{\mathrm{T}}\left(k\right)+{\mathbf{E}}_N^{\mathrm{T}}\cdot \mathbf{Q}\left(k\right)\cdot {\mathbf{E}}_N$$
    
• Update step:
  • measurement residual: z̃(k)=y(k)- C(k)· x̂(k, k-1)
  • residual covariance: S(k)=C(k)·P(k,k-1)·C ^T(k)+R(k)
  • Kalman gain: K(k)=P(k,k-1)·C ^T(k)· S ^-1(k)
  • updated state: x̂(k,k)=x̂(k,k-1)+K(k)· z̃(k)
  • updated covariance: P(k,k)=(I - K(k)·C(k))·P(k,k-1)
• The above calculation based on the Kalman filter is known per se from the prior art and thus will not be explained in detail. By using the Kalman filter, the estimation of the state variables is based on estimated variables of all nodes. Hence, when implementing the Kalman filter, the synchronization is done in a common fusion center in which all known variables for the master node MA and all slave nodes SL1 to SLN are collected. This is indicated in the diagram of Fig. 3. According to this diagram, the measured variables TS( M(k)) of the master node and the variables c_n (k), b_n (k) and a_n (k) (n = 1, ..., N) of the slave nodes are sent from each node to the fusion center FC. In the fusion center, the Kalman filter is applied to the variables resulting in estimated state variables ${\hat{x}}_n^{\mathit{in}}\left(k\right),$

  

  R̂_n (k) (n = 1, ..., N), i.e. in an estimation of the master counter value and the RFC value when receiving the synchronization message in a corresponding slave node. Those estimated variables are passed back to the respective slaves which use those variables in order to synchronize their clocks.
• In the following, a second embodiment for estimating the state variables using the so-called sum-product algorithm will be described on the basis of the above defined state transition and observation model, namely based on the following equations: $$R_n\left(k\right)=R_n\left(k-1\right)+{\omega }_n\left(k\right)$$

  

  $$x_n^{\mathrm{in}}\left(k\right)=x_n^{\mathrm{in}}\left(k-1\right)+a_n\left(k\right)\cdot R_n\left(k-1\right)$$

  

  $$\mathit{TS}\left(M\left(k\right)\right)=x_1^{\mathrm{in}}\left(k\right)-c_1\left(k\right)\cdot R_1\left(k\right)+{\nu }_1\left(k\right)$$

  

  $$\begin{array}{cc}{x}_{n-1}^{\mathrm{in}}\left(k\right)+b_{n-1}\left(k\right)\cdot R_{n-1}\left(k\right)=x_n^{\mathrm{in}}\left(k\right)-c_n\left(k\right)\cdot R_n\left(k\right)+{\nu }_n\left(k\right)& \mathrm{for}n>1\end{array}$$

  
• For more compact notation the two hidden state variables are lumped into one hidden vector-valued variable ${\mathbf{x}}_n\left(k\right)={\left[x_n^{\mathrm{in}}\left(k\right),R_n\left(k\right)\right]}^{\mathrm{T}},$

  

  and it is defined accordingly: $$\begin{array}{ccc}{\mathbf{A}}_n\left(k\right)=\left[\begin{array}{cc}1& a_n\left(k\right)\\ 0& 1\end{array}\right],& \mathbf{e}={\left[0,1\right]}^{\mathrm{T}},& {\mathbf{c}}_n\left(k\right)={\left[1,-c_n\left(k\right)\right]}^{\mathrm{T}}\end{array},$$

  

  $${\mathbf{b}}_n\left(k\right)={\left[1,b_n\left(k\right)\right]}^{\mathrm{T}}\mathrm{.}$$

  
• Then the state-space equations become: $$\begin{array}{|c|}\hline {\mathbf{x}}_n\left(k\right)={\mathbf{A}}_n\left(k\right)\cdot {\mathbf{x}}_n\left(k-1\right)+{\omega }_n\left(k\right)\cdot \mathbf{e}\\ \hline\end{array}$$

  

  $$\begin{array}{|c|}\hline \mathit{TS}\left(M\left(k\right)\right)={\mathbf{c}}_1^{\mathbf{T}}\left(k\right)\cdot {\mathbf{x}}_1\left(k\right)+{\nu }_1\left(k\right)\\ \hline\end{array}$$

  

  $$\begin{array}{|c|}\hline \begin{array}{cc}{\mathbf{b}}_{n-1}^{\mathbf{T}}\left(k\right)\cdot {\mathbf{x}}_{n-1}\left(k\right)={\mathbf{c}}_n^{\mathbf{T}}\left(k\right)\cdot {\mathbf{x}}_n\left(k\right)+{\nu }_n\left(k\right)& \mathrm{for}n>1\end{array}\\ \hline\end{array}$$

  
• The random vectors x _n (k) and x _n (k-1) are related via the probability density function p _{ω n (k)} (ω _n (k)) of the noise ω _n (k); the distribution of x _l(k) is given by the probability density function p _{v 1(k)}(ν₁(k)) of the noise v _l(k); and the random vectors x _n (k) and x _n-1(k) are related via the probability density function p _{vn (k)}(ν₁(k)) of the noise v_n (k).
• It is assumed that all noises are Gaussian, i.e., for all n ∈ {1 ,...N}, ${\omega }_n\left(k\right)\sim N\left(0,{\sigma }_{{\omega }_n\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2\right)$

  

  and ${\nu }_n\left(k\right)\sim N\left(0,{\sigma }_{{\nu }_n\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2\right)\mathrm{.}$

  

  
  Then $$p_{{\omega }_n\left(k\right)}\left({\omega }_n\left(k\right)\right)=\frac{1}{\sqrt{2\pi }{\sigma }_{{\omega }_n\left(k\right)}}\exp \left(-\frac{{\left({\omega }_n\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)\right)}^2}{2{\sigma }_{{\omega }_n\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2}\right)$$

  

  and since from (16') $${\omega }_n\left(k\right)={\mathbf{e}}^{\mathrm{T}}\cdot \left({\mathbf{x}}_n\left(k\right)-{\mathbf{A}}_n\left(k\right){\mathbf{x}}_n\left(k-1\right)\right)$$

  

  the following function is defined: $$\begin{array}{|c|}\hline \begin{array}{l}{f}_k^n\left({\mathbf{x}}_n\left(k\right)|{\mathbf{x}}_n\left(k-1\right)\right)=p_{{\omega }_n\left(k\right)}\left({\omega }_n\left(k\right)\right)=p_{{\omega }_n\left(k\right)}\left({\mathbf{e}}^{\mathrm{T}}\left({\mathbf{x}}_n\left(k\right)-{\mathbf{A}}_n\left(k\right){\mathbf{x}}_n\left(k-1\right)\right)\right)\\ =\frac{1}{\sqrt{2\pi }{\sigma }_{{\omega }_n\left(k\right)}}\exp \left(-\frac{{\left({\mathbf{x}}_n\left(k\right)-{\mathbf{A}}_n\left(k\right){\mathbf{x}}_n\left(k-1\right)\right)}^{\mathrm{T}}{\mathbf{ee}}^{\mathrm{T}}\left({\mathbf{x}}_n\left(k\right)-{\mathbf{A}}_n\left(k\right){\mathbf{x}}_n\left(k-1\right)\right)}{2{\sigma }_{{\omega }_n\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2}\right)\end{array}\\ \hline\end{array}$$

  
• Likewise: $$p_{{\nu }_n\left(k\right)}\left({\nu }_n\left(k\right)\right)=\frac{1}{\sqrt{2\pi }{\sigma }_{{\nu }_n\left(k\right)}}\exp \left(-\frac{{\left({\nu }_n\left(k\right)\right)}^2}{2{\sigma }_{{\nu }_n\left(k\right)}^2}\right)$$

  

  and from (17'), (18') $${\nu }_1\left(k\right)=\mathit{TS}\left(M\left(k\right)\right)-{\mathbf{c}}_1^{\mathbf{T}}\left(k\right)\cdot {\mathbf{x}}_1\left(k\right)$$

  

  $$\begin{array}{cc}{\nu }_n\left(k\right)={\mathbf{b}}_{n-1}^{\mathrm{T}}\left(k\right)\cdot {\mathbf{x}}_{n-1}\left(k\right)-{\mathbf{c}}_n^{\mathrm{T}}\left(k\right)\cdot {\mathbf{x}}_n\left(k\right)& \left(\mathrm{for}n>1\right)\end{array}$$

  

  the following functions are defined: $$\begin{array}{|c|}\hline \begin{array}{l}{g}_k^1\left({\mathbf{x}}_1\left(k\right)|\mathit{TS}\left(M\left(k\right)\right)\right)=p_{{\nu }_1\left(k\right)}\left({\nu }_1\left(k\right)\right)=p_{{\nu }_1\left(k\right)}\left(\mathit{TS}\left(M\left(k\right)\right)-{\mathbf{c}}_1^{\mathbf{T}}\left(k\right)\cdot {\mathbf{x}}_1\left(k\right)\right)\\ =\frac{1}{\sqrt{2\pi }{\sigma }_{{\nu }_1\left(k\right)}}\exp \left(-\frac{{\left(\mathit{TS}\left(M\left(k\right)\right)-{\mathbf{c}}_1^{\mathbf{T}}\left(k\right)\cdot {\mathbf{x}}_1\left(k\right)\right)}^2}{2{\sigma }_{{\nu }_1\left(k\right)}^2}\right)\end{array}\\ \hline\end{array}$$

  

  and $$\begin{array}{|c|}\hline \begin{array}{l}{g}_k^n\left({\mathbf{x}}_n\left(k\right)|{\mathbf{x}}_{n-1}\left(k\right)\right)=p_{{\nu }_n\left(k\right)}\left({\nu }_n\left(k\right)\right)=p_{{\nu }_n\left(k\right)}\left({\mathbf{b}}_{n-1}^{\mathrm{T}}\left(k\right)\cdot {\mathbf{x}}_{n-1}\left(k\right)-{\mathbf{c}}_n^{\mathrm{T}}\left(k\right)\cdot {\mathbf{x}}_n\left(k\right)\right)\\ =\frac{1}{\sqrt{2\pi }{\sigma }_{{\nu }_n\left(k\right)}}\exp \left(-\frac{{\left({\mathbf{b}}_{n-1}^{\mathbf{T}}\left(k\right)\cdot {\mathbf{x}}_{n-1}\left(k\right)-{\mathbf{c}}_n^{\mathrm{T}}\left(k\right)\cdot {\mathbf{x}}_n\left(k\right)\right)}^2}{2{\sigma }_{{\nu }_n\left(k\right)}^2}\right)\end{array}\\ \hline\end{array}$$

  
• The above dependencies can be described in a factor graph which is shown in Fig. 4. Factor graphs per se are known from the prior art. In the factor graph of Fig. 4 a circle is called a variable node, and represents the variable(s) labelling the circle. Shaded circles are observed variables, while un-shaded ones are hidden variables. A square is a function node which represents a function labelling that square. A node variable is connected to a function node if the corresponding variable is mentioned in the function. The whole graph represents the factorization of a global function.
• In the following, the factor graph of Fig. 4 will be used to represent the relationships between the variables based on formulas (21'), (25') and (26'). Generally, a factor graph of a function g can be combined with message passing algorithms, such as the sum-product algorithm, to efficiently compute certain characteristics of g, such as marginals.
• In the embodiment described herein, the sum-product algorithm is used to calculate the posterior probability distribution of x _n (k) given the observation TS(M(k)), which is denoted in the following by q(x _n (k)) and defined in (27') below. A MAP (maximum a posteriori) estimation of the hidden state variables contained in vector x _n (k), can then be obtained from q(x _n (k)).
• A simplified version of the sum-product algorithm in which messages are only passed in one direction is used in the embodiment described herein. Figs. 5A, 5B and 5C show parts of the corresponding factor graph for this simplified algorithm. Particularly, those figures indicate the message propagation on the graph in the sum-product algorithm. Fig. 5A shows the part of the factor graph for variable x ₁(k), Fig. 5B shows the part of the graph for variable x ₂(k) and Fig. 5C shows in general the part of the graph for variable x _n (k). In figures 5A to 5C, solid lines represent messages used within a time step, while dashed lines represent messages passed to the next time step (i.e. the next synchronization cycle).
• Let q(x _n (k)) denote the posterior probability density function of the state x _n (k) obtained in the time step k for slave n, given the observation TS(M(k)). Since all the equations in the state-space model are linear and all the random variables have Gaussian distribution, q(x _n (k)) is also a Gaussian distribution, which is given by: $$q\left({\mathbf{x}}_n\left(k\right)\right)=\frac{1}{\left(2\pi \right){\left|{\mathbf{<figure-callout\; id="1"\; label="P\; n\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">P</figure-callout>}}_n\left(k\right)\left(\right)\right|}^{1/2}}\exp \left(-\frac{1}{2}{\left({\mathbf{x}}_n\left(k\right)-{\bar{\mathbf{x}}}_n\left(k\right)\right)}^{\mathrm{T}}{\mathbf{P}}_n{\left(k\right)}^{-1}\left({\mathbf{x}}_n\left(k\right)-{\bar{\mathbf{x}}}_n\left(k\right)\right)\right)$$

  

  where x̅(k)=E{x _n (k)} is the mean of Gaussian random variable x _n (k). P _n (k) is its covariance matrix, defined as: $${\mathbf{P}}_n\left(k\right)=E\left\{\left({\mathbf{x}}_n\left(k\right)-{\bar{\mathbf{x}}}_n\left(k\right)\right){\left({\mathbf{x}}_n\left(k\right)-{\bar{\mathbf{x}}}_n\left(k\right)\right)}^{\mathrm{T}}\right\}\mathrm{.}$$

  
• The mean and covariance matrix are initialized with appropriate values. E.g., E{R_n (k)} is initialized by 1, $E\left\{x_n^{\mathrm{in}}\left(k\right)\right\}$

  

  is initialized by $\mathit{TS}\left(M\left(k\right)\right)+{\displaystyle \sum _{i=1}^n}{c}_i\left(k\right)+{\displaystyle \sum _{i=1}^{n-1}}{b}_i\left(k\right)$

  

  and P _n (k) is initialized by $N\cdot {\sigma }_{{\nu }_n\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2\cdot {\mathbf{I}}_N$

  

  in step 0. The following part derives the transition from step k-1 to step k.
• Using the sum-product algorithm, q(x _n (k)) can be calculated in each time step k, for which it is sufficient to calculate the mean and variance. Because q(x _n (k)) is Gaussian, it has its maximum at the mean, hence the MAP estimate of x _n (k) is: ${\hat{\mathbf{x}}}_n^{\mathrm{MAP}}\left(k\right)={\bar{\mathbf{x}}}_n\left(k\right)\mathrm{.}$

  
• The above mentioned Figs. 5A to 5C illustrate the sum-product algorithm. If there is an arrow from node a to node b, then a is a parent of b and b is a child of a. A parent node can pass a "message" to a child node, in the direction indicated by the arrows on the edges. It can be observed in Figs. 5A to 5C that each node has at most one child. The messages are real numbers, and namely the values of certain relevant probability density functions. Let m _a→b denote the message from node a to node b.
• The message calculation at each node is carried out according to the sum-product algorithm as follows:
• A node can calculate the message to its child if it has received messages from all parents.
• The message from function node a to variable node i is given by: $$m_{a\to i}\left(y\right)={\displaystyle \underset{x}{\int }}\left(\underset{j\in \mathrm{Pa}\left(a\right)}{\mathrm{\Pi }}{m}_{j\to a}\left(x\right)\right)\cdot f_a\left(x,y\right)\mathit{dx}$$
  
  
  where f_a (x,y) is the function represented by node a and Pa(a) denotes all the parents of a. I.e., all the messages received from the parents are multiplied, then multiplied by the own function, then the desired marginal of this expression is computed (by integration over all other variables).
• The message from a variable node j to a function node a is given by: $$m_{j\to a}\left(x_j\right)=\underset{b\in \mathrm{Pa}\left(j\right)}{\mathrm{\Pi }}{m}_{b\to j}\left(x_j\right)=p_j\left(x_j\right)$$
  
  i.e. consists of the product of all the messages received from the parents. This message computes the marginal probability density function at every variable node if it has received all messages from its parents.
• In (28') and (29'), if incoming messages are Gaussian, then the outgoing message is also Gaussian. The sum-product algorithm will be initialized by Gaussian distributions, so all the messages are Gaussian. µ _a→b and Λ _a→b are used to denote the mean and the variance of message m _a→b, see (33').
• According to (28'), the message from function node $f_k^n$

  

  to variable node x _n (k) can be calculated as follows (see "left arm" in subgraphs of Fig. 5C): $$m_{f_k^n\to {\mathbf{x}}_n\left(k\right)}\left({\mathbf{x}}_n\left(k\right)\right)={\displaystyle {\int }_{{\mathbf{x}}_n\left(k-1\right)}}q\left({\mathbf{x}}_n\left(k-1\right)\right)\cdot f_k^n\left({\mathbf{x}}_n\left(k\right)|{\mathbf{x}}_n\left(k-1\right)\right)$$

  
• The expression of q( x _n (k-1)) is given by (27') and the expression of $f_k^n\left({\mathbf{x}}_n\left(k\right)|{\mathbf{x}}_n\left(k-1\right)\right)$

  

  is given by (21'). Calculating the integral in (30') according to the standard multiplication and integration of multivariate Gaussian distribution functions, leads to the result that $m_{f_k^n\to {\mathbf{x}}_n\left(k\right)}\left({\mathbf{x}}_n\left(k\right)\right)$

  

  is a Gaussian density function, with mean: $${\mu }_{f_k^n\to x_n\left(k\right)}={\mathbf{A}}_n\left(k\right)\cdot {\bar{\mathbf{x}}}_n\left(k-1\right)$$

  

  and covariance matrix: $${\mathbf{\Lambda }}_{f_k^n\to x_n\left(k\right)}={\mathbf{A}}_n\left(k\right){\mathbf{P}}_n\left(k-1\right){\mathbf{A}}_n^{\mathrm{T}}\left(k\right)+{\mathbf{\Phi }}_n\left(k\right)$$

  

  where $${\mathbf{\Phi }}_n\left(k\right)=\left[\begin{array}{cc}0& 0\\ 0& {\sigma }_{{\omega }_n\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2\end{array}\right]\mathrm{.}$$

  
• So $m_{f_k^n\to {\mathbf{x}}_n\left(k\right)}\left({\mathbf{x}}_n\left(k\right)\right)$

  

  can be expressed as: $$\begin{array}{l}{m}_{f_k^n\to {\mathbf{x}}_n\left(k\right)}\left({\mathbf{x}}_n\left(k\right)\right)=\\ =\frac{1}{2\pi \left|{\mathrm{\Lambda }}_{f_k^n\to {\mathbf{x}}_n\left(k\right)}\right|}\exp \left(-\frac{1}{2}{\left({\mathbf{x}}_n\left(k\right)-{\mu }_{f_k^n\to {\mathbf{x}}_n\left(k\right)}\right)}^{\mathrm{T}}{\mathrm{\Lambda }}_{f_k^n\to {\mathbf{x}}_n\left(k\right)}^{-1}\left({\mathbf{x}}_n\left(k\right)-{\mu }_{f_k^n\to {\mathbf{x}}_n\left(k\right)}\right)\right)\end{array}$$

  
• In the following, the message calculation around node x _l(k) will be described. As shown in Fig. 5A, the message from function node $g_k^{}$ ${}_1^{}$

  

  to variable node x _l(k), i.e. $m_{{\mathrm{<figure-callout\; id="1"\; label="g\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}\mathrm{\to }{\mathbf{x}}_1\left(k\right)},$

  

  is only determined by the function $g_k^{}$ ${}_1^{},$

  

  since no message enters this function node. According to the sum-product algorithm, $m_{{\mathrm{<figure-callout\; id="1"\; label="g\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}\mathrm{\to }{\mathbf{x}}_1\left(k\right)}$

  

  is given by: $$m_{{\mathrm{<figure-callout\; id="1"\; label="g\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}\mathrm{\to }{\mathbf{x}}_1\left(k\right)}\left({\mathbf{x}}_1\left(k\right)\right)=g_k^1\left({\mathbf{x}}_1\left(k\right)\right)$$

  
• As $g_k^1\left({\mathbf{x}}_1\left(k\right)|\mathit{TS}\left(M\left(k\right)\right)\right)$

  

  is given by (25'), the expression of $m_{{\mathrm{<figure-callout\; id="1"\; label="g\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}\mathrm{\to }{\mathbf{x}}_1\left(k\right)}\left({\mathbf{x}}_1\left(k\right)\right)$

  

  is identical to (25'), namely: $$m_{{\mathrm{<figure-callout\; id="1"\; label="g\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}\mathrm{\to }{\mathbf{x}}_1\left(k\right)}^1\left({\mathbf{x}}_1\left(k\right)\right)=\frac{1}{\sqrt{2\pi }{\sigma }_{{\nu }_1\left(k\right)}}\exp \left(-\frac{{\left(\mathit{TS}\left(M\left(k\right)\right)-{\mathbf{c}}_1^{\mathbf{T}}\left(k\right)\cdot {\mathbf{x}}_1\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)\right)}^2}{2{\sigma }_{{\nu }_1\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2}\right)$$

  
• According to Figure 5A, the only incoming messages at variable node x _l(k) are $m_{{\mathrm{<figure-callout\; id="1"\; label="f\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">f</figure-callout>}}_k^{}\mathrm{\to }{\mathbf{x}}_1\left(k\right)}\left({\mathbf{x}}_1\left(k\right)\right)$

  

  and $m_{{\mathrm{<figure-callout\; id="1"\; label="g\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}\mathrm{\to }{\mathbf{x}}_1\left(k\right)}\left({\mathbf{x}}_1\left(k\right)\right)\mathrm{.}$

  

  Since now they have been calculated in (30') and (34'), and the results are given in (33') and (35'), the estimated marginal q(x ₁(k)) according to (29') is calculated as follows: $$q\left({\mathbf{x}}_1\left(k\right)\right)=m_{{\mathrm{<figure-callout\; id="1"\; label="g\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}\mathrm{\to }{\mathbf{x}}_1\left(k\right)}\left({\mathbf{x}}_1\left(k\right)\right)\cdot m_{{\mathrm{<figure-callout\; id="1"\; label="f\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">f</figure-callout>}}_k^{}\mathrm{\to }{\mathbf{x}}_1\left(k\right)}\left({\mathbf{x}}_1\left(k\right)\right)$$

  
• Calculating the product of two Gaussian functions in (36'), it is found out that q(x ₁(k)) is also a Gaussian function. Its covariance matrix is: $${\mathbf{P}}_1\left(k\right)={\left({\mathbf{c}}_1\left(k\right){\sigma }_{{\nu }_1}^{-2}{\mathbf{c}}_1^{\mathrm{T}}\left(k\right)+{\mathbf{<figure-callout\; id="1"\; label="\Lambda \; f\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">\Lambda </figure-callout>}}_{f_k^{}}^{{}_1^{}\to {\mathbf{x}}_1\left(k\right)}\right)}^{-1}$$

  

  and the mean is: $${\mathbf{x}}_1\left(k\right)={\mathbf{P}}_1\left(k\right)\cdot \left({\mathbf{c}}_1\left(k\right){\sigma }_{{\nu }_1}^{-2}\mathit{TS}\left(M\left(k\right)\right)+{\mathbf{<figure-callout\; id="1"\; label="\Lambda \; f\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">\Lambda </figure-callout>}}_{f_k^{}}^{{}_1^{}\to {\mathbf{x}}_1\left(k\right)}{\mu }_{{\mathrm{<figure-callout\; id="1"\; label="f\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">f</figure-callout>}}_k^{}\to {\mathbf{x}}_1\left(k\right)}\right)$$

  
• From Figure 5A, it can be seen that the message that variable node x ₁(k) sends to function node $g_k^{}$ ${}_2^{}$

  

  should be calculated by, according to (29'): $$m_{{\mathbf{x}}_1\left(k\right)\mathrm{\to }{g}_k^2}\left({\mathbf{x}}_1\left(k\right)\right)=m_{g_k^1\left(k\right)\mathrm{\to }{\mathbf{x}}_1\left(k\right)}\left({\mathbf{x}}_1\left(k\right)\right)\cdot m_{f_k^1\left(k\right)\mathrm{\to }{\mathbf{x}}_1\left(k\right)}\left({\mathbf{x}}_1\left(k\right)\right)$$

  
• Comparing (39') and (36'), it can be seen that: $$m_{{\mathbf{x}}_1\left(k\right)\to g_k^2}\left({\mathbf{x}}_1\left(k\right)\right)=q\left({\mathbf{x}}_1\left(k\right)\right)$$

  
• So $m_{{\mathbf{x}}_1\left(k\right)\to g_k^2}\left({\mathbf{x}}_1\left(k\right)\right)$

  

  is also a Gaussian density function. Its mean and covariance matrix are given by: $${\mathbf{\Lambda }}_{{\mathbf{x}}_1\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}}={\left({\mathbf{c}}_1\left(k\right){\sigma }_{{\nu }_1}^{-2}{\mathbf{c}}_1^{\mathbf{T}}\left(k\right)+{\mathbf{<figure-callout\; id="1"\; label="\Lambda \; f\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">\Lambda </figure-callout>}}_{f_k^{}}^{{}_1^{}\to {\mathbf{x}}_1\left(k\right)}\right)}^{-1}$$

  

  and $${\mathbf{\mu }}_{{\mathbf{x}}_1\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}}={\mathbf{\Lambda }}_{{\mathbf{x}}_1\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}}{\left({\mathbf{c}}_1\left(k\right){\sigma }_{{\nu }_1}^{-2}\mathit{TS}\left(M\left(k\right)\right)+{\mathbf{<figure-callout\; id="1"\; label="\Lambda \; f\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">\Lambda </figure-callout>}}_{f_k^{}}^{{}_1^{}\to {\mathbf{x}}_1\left(k\right)}{\mathbf{\mu }}_{{\mathrm{<figure-callout\; id="1"\; label="f\; k"\; filenames="imgf0003.png,imgf0004.png"\; state="\{\{state\}\}">f</figure-callout>}}_k^{}\to {\mathbf{x}}_1\left(k\right)}\right)}^{-1}$$

  
• So $m_{{\mathbf{x}}_1\left(k\right)\mathrm{\to }{g}_k^2}\left({\mathbf{x}}_1\left(k\right)\right)$

  

  is expressed by: $$\begin{array}{l}{m}_{{\mathbf{x}}_1\left(k\right)}\left({\mathbf{x}}_1\left(k\right)\right)=\\ =\frac{1}{2\pi \left|{\mathbf{\Lambda }}_{{\mathbf{x}}_1\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}}\right|}\exp \left(-\frac{1}{2}{\left({\mathbf{x}}_n\left(k\right)-{\mathbf{\mu }}_{{\mathbf{x}}_1\left(k\right)\to g_k^2}\right)}^T{\mathbf{\Lambda }}_{{\mathbf{x}}_1\left(k\right)\to g_k^2}^{-1}\left({\mathbf{x}}_n\left(k\right)-{\mathbf{\mu }}_{{\mathbf{x}}_1\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}}\right)\right)\end{array}$$

  
• Now the message calculation around node x ₁(k) based on Fig. 5A is finished. The calculation generates the marginal probability density q(x _l(k)) and the message goes to node x ₂(k), i.e. $m_{{\mathbf{x}}_1\left(k\right)\to g_k^2}\left({\mathbf{x}}_1\left(k\right)\right)\mathrm{.}$

  
• In the following, the message calculation around node x ₂(k) is described. Having received $m_{{\mathbf{x}}_1\left(k\right)\to g_k^2}\left({\mathbf{x}}_1\left(k\right)\right)$

  

  from x ₁(k), ${\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}$

  

  can calculate its message to x ₂(k) according to Fig. 5B by using (28'): $$m_{{\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}\to {\mathbf{x}}_2\left(k\right)}\left({\mathbf{x}}_2\left(k\right)\right)={\int }_{{\mathbf{x}}_1\left(k\right)}{m}_{{\mathbf{x}}_1\left(k\right)\to g_k^2}\left({\mathbf{x}}_1\left(k\right)\right)\cdot g_k^2\left({\mathbf{x}}_2\left(k\right)|{\mathbf{x}}_1\left(k\right)\right)$$

  

  where $m_{{\mathbf{x}}_1\left(k\right)\to g_k^2}\left({\mathbf{x}}_1\left(k\right)\right)$

  

  is given by (43') and $g_k^2\left({\mathbf{x}}_2\left(k\right)|{\mathbf{x}}_1\left(k\right)\right)$

  

  is given by (26'). The result of (44') is given by: $$m_{{\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}\to {\mathbf{x}}_2\left(k\right)}\left({\mathbf{x}}_2\left(k\right)\right)=\alpha \cdot \exp \left(-\frac{{\left({\mathbf{c}}_2^{\mathrm{T}}\left(k\right){\mathbf{x}}_2\left(k\right)-{\mathbf{b}}_1^{\mathrm{T}}\left(k\right){\mathbf{\mu }}_{{\mathbf{x}}_1\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}}\right)}^2}{2\left({\sigma }_{{\nu }_2\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2+{\mathbf{b}}_1^{\mathbf{T}}\left(k\right){\mathbf{\Lambda }}_{{\mathbf{x}}_1\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}}{\mathbf{b}}_1\left(k\right)\right)}\right)$$

  

  where α is normalization factor, ${\mathbf{\mu }}_{{\mathbf{x}}_1\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}}$

  

  is given by (42') and ${\mathbf{\Lambda }}_{{\mathbf{x}}_1\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}}$

  

  is given by (41').
• According to Figure 5B, the only incoming messages at variable node x ₂(k) are $m_{f_k^2\left(k\right)\mathrm{\to }{\mathbf{x}}_2\left(k\right)}\left({\mathbf{x}}_2\left(k\right)\right)$

  

  and $m_{g_k^2\left(k\right)\mathrm{\to }{\mathbf{x}}_2\left(k\right)}\left({\mathbf{x}}_2\left(k\right)\right)\mathrm{.}$

  

  Since now they have been calculated in (30') and (44'), and the results are given in (33') and (45'), the estimated marginal q(x ₂(k) can be calculated according to (29) as follows: $$q\left({\mathbf{x}}_2\left(k\right)\right)=m_{{\mathrm{<figure-callout\; id="2"\; label="f\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">f</figure-callout>}}_k^{}\mathrm{\to }{\mathbf{x}}_2\left(k\right)}\left({\mathbf{x}}_2\left(k\right)\right)\cdot m_{{\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}\mathrm{\to }{\mathbf{x}}_2\left(k\right)}\left({\mathbf{x}}_2\left(k\right)\right)$$

  
• Inserting (33') and (45') in (46'), the result of q(x ₂(k)) is a Gaussian distribution. Its covariance matrix and mean are given by: $${\mathbf{P}}_2\left(k\right)={\left({\mathbf{c}}_2\left(k\right){\left[{\sigma }_{{\nu }_2\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2+{\mathbf{b}}_1^{\mathbf{T}}\left(k\right){\mathbf{\Lambda }}_{{\mathbf{x}}_1\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}}{\mathbf{b}}_1\left(k\right)\right]}^{-1}{\mathbf{c}}_2^{\mathbf{T}}\left(k\right)+{\mathbf{<figure-callout\; id="2"\; label="\Lambda \; f\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">\Lambda </figure-callout>}}_{f_k^{}}^{{}_2^{}\to {\mathbf{x}}_2\left(k\right)}\right)}^{-1}$$

  

  and $${\bar{\mathbf{x}}}_2\left(k\right)={\mathbf{P}}_2\left(k\right)\left\{\begin{array}{c}{\mathbf{c}}_2\left(k\right){\left[{\sigma }_{{\nu }_2\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2+{\mathbf{b}}_1^{\mathrm{T}}\left(k\right){\mathbf{\Lambda }}_{{\mathbf{x}}_1\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}}{\mathbf{b}}_1\left(k\right)\right]}^{-1}{\mathbf{b}}_1^{\mathrm{T}}\left(k\right){\mathit{\mu }}_{{\mathbf{x}}_1\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}}\\ +{\mathbf{<figure-callout\; id="2"\; label="\Lambda \; f\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">\Lambda </figure-callout>}}_{f_k^{}}^{{}_2^{}\to {\mathbf{x}}_2\left(k\right)}{\mathit{\mu }}_{{\mathrm{<figure-callout\; id="2"\; label="f\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">f</figure-callout>}}_k^{}\to {\mathbf{x}}_2\left(k\right)}\end{array}\right\}$$

  
• From Fig. 5B, it can be seen that message $m_{{{\mathbf{x}}_2\left(k\right)\to g}_k^3}\left({\mathbf{x}}_1\left(k\right)\right)$

  

  should be calculated according to (29') by: $$m_{{{\mathbf{x}}_2\left(k\right)\to g}_k^3}\left({\mathbf{x}}_1\left(k\right)\right)=m_{{\mathrm{<figure-callout\; id="2"\; label="f\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">f</figure-callout>}}_k^{}\to {\mathbf{x}}_2\left(k\right)}\left({\mathbf{x}}_2\left(k\right)\right)\cdot m_{{\mathrm{<figure-callout\; id="2"\; label="g\; k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_k^{}\to {\mathbf{x}}_2\left(k\right)}\left({\mathbf{x}}_2\left(k\right)\right)$$

  

  where $m_{f_k^2\left(k\right)\mathrm{\to }{\mathbf{x}}_2\left(k\right)}\left({\mathbf{x}}_2\left(k\right)\right)$

  

  is given by (33') and $m_{g_k^2\left(k\right)\mathrm{\to }{\mathbf{x}}_2\left(k\right)}\left({\mathbf{x}}_2\left(k\right)\right)$

  

  is given by (45'). The calculation in (49') results in a Gaussian density function. Its mean ${\mathbf{\mu }}_{{\mathbf{x}}_1\left(k\right)\to g_k^2}\left({\mathbf{x}}_1\left(k\right)\right)$

  

  and covariance matrix ${\mathbf{\Lambda }}_{{\mathbf{x}}_1\left(k\right)\to g_k^2}\left({\mathbf{x}}_1\left(k\right)\right)$

  

  are given by: $${\mathbf{\Lambda }}_{{\mathbf{x}}_2\left(k\right)\to g_k^3}\left({\mathbf{x}}_2\left(k\right)\right)={\left({\mathbf{c}}_2\left(k\right){\left[{\sigma }_{{\nu }_2\left(k\right)}^2+{\mathbf{b}}_1^{\mathrm{T}}\left(k\right){\mathbf{\Lambda }}_{{\mathbf{x}}_1\left(k\right)\to g_k^2}{\mathbf{b}}_1\left(k\right)\right]}^{-1}{\mathbf{c}}_2^{\mathrm{T}}\left(k\right)+{\mathbf{\Lambda }}_{f_k^2\to {\mathbf{x}}_2\left(k\right)}^{-1}\right)}^{-1}$$

  

  and $${\mathbf{\mu }}_{{\mathbf{x}}_2\left(k\right)\to g_k^3}\left({\mathbf{x}}_2\left(k\right)\right)={\mathbf{\Lambda }}_{{\mathbf{x}}_2\left(k\right)\to g_k^3}\left({\mathbf{x}}_2\left(k\right)\right)\left\{\begin{array}{c}{\mathbf{c}}_2\left(k\right){\left[{\sigma }_{{\nu }_2\left(k\right)}^2+{\mathbf{b}}_1^{\mathrm{T}}\left(k\right){\mathbf{\Lambda }}_{{\mathbf{x}}_1\left(k\right)\to g_k^2}{\mathbf{b}}_1\left(k\right)\right]}^{-1}{\mathbf{b}}_1^{\mathrm{T}}\left(k\right){\mathit{\mu }}_{{\mathbf{x}}_1\left(k\right)\to g_k^2}\\ +{\mathbf{\Lambda }}_{f_k^2\to {\mathbf{x}}_2\left(k\right)}^{-1}{\mathit{\mu }}_{f_k^2\to {\mathbf{x}}_2\left(k\right)}\end{array}\right\}$$

  

  $m_{{{\mathbf{x}}_2\left(k\right)\to g}_k^3}\left({\mathbf{x}}_2\left(k\right)\right)$

  

  is then given by_: $$\begin{array}{l}{m}_{{\mathbf{x}}_2\left(k\right)\to g_k^3}\left({\mathbf{x}}_2\left(k\right)\right)=\\ =\frac{1}{2\pi \left|{\mathbf{\Lambda }}_{{\mathbf{x}}_2\left(k\right)\to g_k^3}\right|}\exp \left(-\frac{1}{2}{\left({\mathbf{x}}_2\left(k\right)-{\mathbf{\mu }}_{{\mathbf{x}}_2\left(k\right)\to g_k^3}\right)}^T{\mathbf{\Lambda }}_{{\mathbf{x}}_2\left(k\right)\to g_k^3}^{-1}\left({\mathbf{x}}_2\left(k\right)-{\mathbf{\mu }}_{{\mathbf{x}}_2\left(k\right)\to g_k^3}\right)\right)\end{array}$$

  
• Now the message calculation around node x ₂(k) based on Fig. 5B is finished. The calculation generates the marginal probability density q(x ₂(k) and the message goes to node x ₂(k), i.e. $m_{{{\mathbf{x}}_2\left(k\right)\to g}_k^3}\left({\mathbf{x}}_1\left(k\right)\right)\mathrm{.}$

  
• In the following, the previous derivation is generalized for the calculation of any given node x _n (k). I.e., the message calculation around node x _n (k) given the input $m_{{{\mathbf{x}}_{n-1}\left(k\right)\to g}_k^n}\left({\mathbf{x}}_{n-1}\left(k\right)\right)$

  

  will be described.
• Having received $m_{{{\mathbf{x}}_{n-1}\left(k\right)\to g}_k^n}\left({\mathbf{x}}_{n-1}\left(k\right)\right)$

  

  from x _n-1(k), $g_k^n$

  

  can calculate its message to x _n (k) according to Figure 5C by using (28'): $$m_{g_k^n\to {\mathbf{x}}_n\left(k\right)}\left({\mathbf{x}}_n\left(k\right)\right)={\displaystyle {\int }_{{\mathbf{x}}_{n-1}\left(k\right)}}{m}_{{\mathbf{x}}_{n-1}\left(k\right)\to g_k^n}\left({\mathbf{x}}_{n-1}\left(k\right)\right)\cdot g_k^n\left({\mathbf{x}}_n\left(k\right),{\mathbf{x}}_{n-1}\left(k\right)\right)$$

  

  $$m_{g_k^n\to {\mathbf{x}}_n\left(k\right)}\left({\mathbf{x}}_n\left(k\right)\right)={\displaystyle {\int }_{{\mathbf{x}}_{n-1}\left(k\right)}}{m}_{{\mathbf{x}}_{n-1}\left(k\right)\to g_k^n}\left({\mathbf{x}}_{n-1}\left(k\right)\right)\cdot g_k^n\left({\mathbf{x}}_n\left(k\right)|{\mathbf{x}}_{n-1}\left(k\right)\right)$$

  

  where $g_k^n\left({\mathbf{x}}_n\left(k\right)|{\mathbf{x}}_{n-1}\left(k\right)\right)$

  

  is given by (26') and $m_{{{\mathbf{x}}_{n-1}\left(k\right)\to g}_k^n}\left({\mathbf{x}}_{n-1}\left(k\right)\right)$

  

  is from the calculation in the previous node, i.e. node x _n-1(k). From the previous derivation it can be seen that $m_{{{\mathbf{x}}_{n-1}\left(k\right)\to g}_k^n}\left({\mathbf{x}}_{n-1}\left(k\right)\right)$

  

  is a Gaussian density function. Its mean will be denoted by ${\mathbf{\mu }}_{{\mathbf{x}}_{n-1}\left(k\right)\to g_k^n}$

  

  and its covariance matrix by ${\mathbf{\Lambda }}_{{\mathbf{x}}_{n-1}\left(k\right)\to g_k^n}\mathrm{.}$

  
• As a consequence, the result of (53') is given by: $$m_{g_k^n\to {\mathbf{x}}_n\left(k\right)}\left({\mathbf{x}}_n\left(k\right)\right)=\alpha \cdot \exp \left(-\frac{{\left({\mathbf{c}}_n^{\mathrm{T}}\left(k\right){\mathbf{x}}_n\left(k\right)-{\mathbf{b}}_{n-1}^{\mathrm{T}}\left(k\right){\mathbf{\mu }}_{{\mathbf{x}}_{n-1}\left(k\right)\to {\mathrm{<figure-callout\; id="2"\; label="g\; k\; n"\; filenames="imgf0002.png"\; state="\{\{state\}\}">g</figure-callout>}}_{kn}^{}}\right)}^2}{2\left({\sigma }_{{\nu }_n\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2+{\mathbf{b}}_{n-1}^{\mathrm{T}}\left(k\right){\mathbf{\Lambda }}_{{\mathbf{x}}_{n-1}\left(k\right)\to g_k^n}{\mathbf{b}}_{i-1}\left(k\right)\right)}\right)$$

  

  where α is the normalization factor.
• Then each q(x _n (k)) is calculated by: $$q\left({\mathbf{x}}_n\left(k\right)\right)=m_{f_k^n\to {\mathbf{x}}_n\left(k\right)}\left({\mathbf{x}}_n\left(k\right)\right)\cdot m_{g_k^n\to {\mathbf{x}}_n\left(k\right)}\left({\mathbf{x}}_n\left(k\right)\right)$$

  
• Inserting the result of (30') and (54') into (55'), it can be found out that the covariance matrix of q(x _n (k)) is: $${\mathbf{P}}_n\left(k\right)={\left({\mathbf{c}}_n\left(k\right){\left[{\sigma }_{{\nu }_n\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2+{\mathbf{b}}_{n-1}^{\mathrm{T}}\left(k\right){\mathbf{\Lambda }}_{{\mathbf{x}}_{n-1}\left(k\right)\to g_k^n}{\mathbf{b}}_{n-1}\left(k\right)\right]}^{-1}{\mathbf{c}}_n^{\mathrm{T}}\left(k\right)+{\mathbf{\Lambda }}_{f_k^n\to {\mathbf{x}}_i\left(k\right)}^{-1}\right)}^{-1}$$

  
• The mean of q(x _n (k)) is: $${\mathbf{x}}_n\left(k\right)={\mathbf{P}}_n\left(k,k\right)\left\{\begin{array}{c}{\mathbf{c}}_n\left(k\right){\left[{\sigma }_{{\nu }_n\left(\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="imgf0002.png"\; state="\{\{state\}\}">k</figure-callout>}\right)}^2+{\mathbf{b}}_{n-1}^{\mathrm{T}}\left(k\right){\mathbf{\Lambda }}_{{\mathbf{x}}_{n-1}\left(k\right)\to g_k^n}{\mathbf{b}}_{n-1}\left(k\right)\right]}^{-1}{\mathbf{b}}_{n-1}^{\mathrm{T}}\left(k\right){\mathbf{\mu }}_{{\mathbf{x}}_{n-1}\left(k\right)\to g_k^n}\\ +{\mathbf{\Lambda }}_{f_k^n\to {\mathbf{x}}_i\left(k\right)}^{-1}{\mathbf{\mu }}_{f_k^n\to {\mathbf{x}}_i\left(k\right)}\end{array}\right\}$$

  
• And the message from variable node x _n (k) to function node $g_k^{n+1}$

  

  is given by: $$m_{{\mathbf{x}}_n\left(k\right)\to g_k^{n+1}}\left({\mathbf{x}}_n\left(k\right)\right)=m_{f_k^n\to {\mathbf{x}}_n\left(k\right)}\left({\mathbf{x}}_n\left(k\right)\right)\cdot m_{g_k^n\to {\mathbf{x}}_n\left(k\right)}\left({\mathbf{x}}_n\left(k\right)\right)=q^s\left({\mathbf{x}}_n\left(k\right)\right)$$

  
• So $m_{{\mathbf{x}}_n\left(k\right)\to g_k^{n+1}}\left({\mathbf{x}}_n\left(k\right)\right)$

  

  is a Gaussian density function and its mean ${\mathbf{\mu }}_{{\mathbf{x}}_n\left(k\right)\to g_k^{n+1}}$

  

  and variance ${\mathbf{\Lambda }}_{{\mathbf{x}}_n\left(k\right)\to g_k^{n+1}}$

  

  are given by: $${\mathbf{\Lambda }}_{{\mathbf{x}}_n\left(k\right)\to g_k^{n+1}}={\mathbf{P}}_n\left(k\right)$$

  

  and $${\mathbf{\mu }}_{{\mathbf{x}}_n\left(k\right)\to g_k^{n+1}}={\bar{\mathbf{x}}}_n\left(k,k\right)$$

  
• Using (53') to (60'), the posterior probability density function for all variable nodes can be calculated.
• The advantage of the above described sum-product algorithm in comparison to the Kalman filter lies in the fact that no fusion center is needed for estimating the hidden state variables. This is because the estimated variable x _n (k), i.e. its mean x̅ _n (k) and its covariance matrix P _n (k), in a slave node n only depends on variables calculated or measured in the previous slave node n-1, i.e. on the previously estimated hidden state variable x _n-1(k) and on the bridge delay b _n-1(k) of the slave node n-1. For the first slave node 1, the estimated hidden state variable only depends on the measured variable TS(M(k)). As a consequence, the synchronization messages can pass the relevant variables for calculating the hidden state variables from one node to another. This is illustrated in Fig. 6. In this figure, the relevant information carried from a master node MA to the slave node SL1, from the slave node SL1 to the slave node SL2 and from the slave node SL2 to the next slave node are indicated next to corresponding arrows illustrating the transmission of synchronization messages between the nodes. Evidently, by using such synchronization messages, each node can calculate the hidden state variable by its own and the variables need not be transmitted to a fusion center in which the calculation is done after collecting known variables from all nodes. Hence, when using the sum-product algorithm, the estimation of the hidden state variable can be performed in less time, thus resulting in a faster synchronization of the clocks.

Claims (19)

1. A method for clock synchronization in a communication network comprising a plurality of nodes (MA, SL1, SL2), wherein the nodes (MA, SL1, SL2) comprise a master node (MA) and slave nodes (SL1, SL2), wherein the master node (MA) has a master clock generating master counter values based on a master counter with a master frequency and wherein each slave node (SL1, SL2) has a slave clock generating slave counter values based on a slave counter with a corresponding slave frequency, the clock synchronization being performed in successive synchronization cycles, wherein in one synchronization cycle:

   - synchronization messages (SY(k)) are transmitted in a sequence of synchronization messages (SY(k)) from the master node (MA) to a slave node (SL1) and from one slave node (SL1) to another slave node (SL2), wherein a synchronization message (SY(k)) sent from one node (MA, SL1, SL2) to another node (MA, SL1, SL2) includes information based on one or more measured counter values (TS(M(k)), TS(S₁ ⁱⁿ(k)), TS(S₁ ^out(k))) of the node (MA, SL1, SL2) sending the synchronization message (SY(k));

   - for each slave node (SL1, SL2) the master counter value (x₁ ⁱⁿ(k), X₂ ⁱⁿ(k)) for a measured slave counter value (TS(S₁ ⁱⁿ(k)), TS(S₂ ⁱⁿ(k))) is estimated by using an estimation method applied to a probabilistic model for state variables of the nodes (MA, SL1, SL2), wherein each slave clock is synchronized based on the estimated master counter value (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k)), wherein the probabilistic model includes.

   - a state transition model describing the dependency of state variables (x₁ ⁱⁿ(k), X₂ ⁱⁿ(k), R₁(k), R₂(k)) of slave nodes (SL1, SL2) in a synchronization cycle on state variables of slave nodes (SL1, SL2) in the previous synchronization cycle;

   - an observation model describing the dependency of known variables (TS(M(k)), b_n(k), c_n(k)) of master and slave nodes (MA, SL1, SL2) on state variables (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k), R₁(k), R₂(k)) of slave nodes (SL1, SL2) ;

   and wherein the state variables (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k), R₁(k), R₂(k)) of the slave nodes (MA, SL1, SL2) are estimated by the estimation method,
   characterized in that the estimation method is based on a Kalman-Filter, in which the estimation of the state variables is based on variables of all nodes, wherein in each
   synchronization cycle the known variables (TS(M(k)), b_n(k), c_n(k)) of each node (MA, SL1, SL2) are transmitted to a common fusion center (FC) in which based on the state transition model and the observation model the state variables (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k), R₁(k), R₂(k)) of each slave node (MA, SL1, SL2) are estimated by the estimation method, said estimated state variables (X₁ ⁱⁿ(k), x₂ ⁱⁿ(k), R₁(k), R₂(k)) being transmitted to the corresponding nodes (MA, SL1, SL2) in order to synchronize the slave clocks.
2. The method according to claim 1, wherein a synchronization message (SY(k)) sent from the master node (MA) includes the measured master counter value (TS(M(k)) when the synchronization message (SY(k) is sent from the master node (MA).
3. The method according to claim 1 or 2, wherein a synchronization message (SY(k)) sent from a slave node (SL1, SL2) includes an approximate master counter value when the synchronization message (SY(k)) is sent from the slave node (SL1, SL2), said approximate master counter value being derived from measured slave counter values (TS(S₁ ⁱⁿ(k)), TS(S₁ ^out(k))) of the slave node (SL1, SL2) and from information received in synchronization messages.
4. The method according to one of the preceding claims, wherein a synchronization message (SY(k)) sent from a slave node (SL1, SL2) includes a measured bridge delay (b₁(k), b₂(k)) between receiving the previous synchronization message and sending the synchronization message (SY(K)) as well as the estimated master counter value (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k)) for the slave node (SL1, SL2).
5. The method according to one of the preceding claims, wherein the probabilistic model incorporates measurement errors with respect to measured counter values (TS(M(k), TS(S₁ ⁱⁿ(k), TS(S₁ ^out(k)) as well as variations of the frequency ratio ((R₁(k), R₂(k)) between the master clock and each slave clock.
6. The method according to one of the preceding claims, wherein the state variables (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k), R₁(k), R₂(k)) comprise the master counter value (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k)) and the frequency ratio (R₁(k), R₂(k)) between the master frequency and a slave frequency when the synchronization message (SY(k)) is received in a slave node.
7. the method according to claim 6, wherein the state transition model is defined as follows: $$R_n\left(k\right)=R_n\left(k-1\right)+{\omega }_n\left(k\right)$$

   

   $$x_n^{\mathrm{in}}\left(k\right)=x_n^{\mathrm{in}}\left(k-1\right)+a_n\left(k\right)\cdot R_n\left(k-1\right),$$

   

   
   where n refers to the nth slave node (SL1, SL2) receiving a synchronization message (SY(k)) in the sequence of synchronization messages (SY(k));
   where k refers to the kth synchronization cycle;
   where $x_n^{\mathrm{in}}\left(k\right)$

   

   refers to the master counter value at the time of receiving the synchronization message (SY(k)) in the nth slave node (SL1, SL2) in the kth synchronization cycle;
   where R_n (k) is the frequency ratio between the master frequency and the slave frequency of the nth slave node (SL1, SL2) in the kth synchronization cycle;
   where ω _n (k) refers to a random variable modelling frequency variations;
   where a_n (k) is the time interval measured in slave counter values between receiving the synchronization message (SY(k)) in the nth slave node (SL1, SL2) in the kth synchronization cycle and receiving the corresponding synchronization message (SY(k-1)) in the nth slave node (SL1, SL2) in the (k-1)th synchronization cycle.
8. The method according to claim 7, wherein the random variable ω _n (k) is modelled as a Gaussian distribution.
9. The method according to one of the preceding claims, wherein the known variables (TS(M(k), b_n(k), c_n(k)) comprise a separately estimated line delay (c_n(k)) in slave counter values between receiving successive synchronization messages (SY(k)) in a slave node (SL1, SL2), a measured bridge delay (b_n(k)) in slave counter values between receiving a synchronization message (SY(k)) and sending the next synchronization message in a slave node (SL1, SL2) and the measured master counter value (TS(M(k)) when the master node (MA) sends a synchronization message (SY(k)).
10. The method according to claim 9, wherein the observation model is defined as follows: $$\mathit{TS}\left(M\left(k\right)\right)=x_1^{\mathrm{in}}\left(k\right)-c_1\left(k\right)\cdot R_1\left(k\right)+{\nu }_1\left(k\right)$$

    

    $$\begin{array}{cc}{x}_{n-1}^{\mathrm{in}}\left(k\right)+b_{n-1}\left(k\right)\cdot R_{n-1}\left(k\right)=x_n^{\mathrm{in}}\left(k\right)-c_n\left(k\right)\cdot R_n\left(k\right)+{\nu }_n\left(k\right)& \mathrm{for}n>1\end{array}$$

    

    
    where n refers to the nth slave node (SL1, SL2) receiving a synchronization message (SY(k)) in the sequence of synchronization messages (SY(k));
    where k refers to the kth synchronization cycle;
    where TS(M(k)) refers to the measured master counter value when the master node (MA) sends the synchronization message (SY(k)) in the kth synchronization cycle;
    where $x_n^{\mathrm{in}}\left(k\right)$

    

    refers to the master counter value at the time of receiving the synchronization message in the nth slave node in the kth synchronization cycle;
    where R _n (k) is the frequency ratio between the master frequency and the slave frequency of the nth slave node (SL1, SL2) in the kth synchronization cycle;
    where c_n (k) refers to the separately estimated line delay in slave counter values between receiving successive synchronization messages (SY(k)) in the nth slave node in the kth synchronization cycle;
    where b_n (k) refers to the measured bridge delay in slave counter values between receiving the synchronization message (SY(k)) and sending the next synchronization message (SY(k)) in the nth slave node in the kth synchronization cycle;
    where v_n (k) refers to a random variable modelling measurement errors in the nth slave node.
11. The method according to claim 10, wherein the random variable ν _n (k) is modelled by a Gaussian distribution, said Gaussian distribution preferably being generated from test sample data.
12. The method according to one of the preceding claims, wherein the state variables (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k), R₁(k), R₂(k)) of a slave node (SL1, SL2) are estimated in the slave node (SL1, SL2).
13. The method according to one of the preceding claims, wherein the estimation method is based on the sum-product algorithm applied to a factor graph describing the relationships between known variables (TS(M(k), b_n(k), c_n(k)) and state variables (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k), R₁(k), R₂(k)) based on message flows.
14. The method according to claim 13, wherein the factor graph has constraints with respect to the message flows such that messages are not passed from a node (MA, SL1, SL2) receiving a synchronising message (SY(k)) to a node from which the synchronising message (SY(k)) is received and such that messages are not passed back in time.
15. The method according to one of the preceding claims, wherein the clock synchronization is based on the standard IEEE 1588.
16. The method according to one of the preceding claims, wherein the nodes (MA, SL1, SL2) communicate with each other based on the PROFINET standard.
17. The method according to one of the preceding claims, wherein the method is used in an industrial automation system.
18. Communication network comprising a plurality of nodes (MA, SL1, SL2), wherein the nodes (MA, SL1, SL2) comprise a master node (MA) and slave nodes (SL1, SL2), wherein the master node (MA) has a master clock generating master counter values based on a master counter with a master frequency and wherein each slave node (SL1, SL2) has a slave clock generating slave counter values based on a slave counter with a corresponding slave frequency, wherein the nodes in the communication network are adapted to perform a clock synchronization in successive synchronization cycles such that in one synchronization cycle:

    - synchronization messages (SY(k)) are transmitted in a sequence of synchronization messages (SY(k)) from the master node (MA) to a slave node (SL1) and from one slave node (SL1) to another slave node (SL2), wherein a synchronization message (SY(k)) sent from one node (MA, SL1, SL2) to another node (MA, SL1, SL2) includes information based on one or more measured counter values (TS(M(k)), TS(S₁ ⁱⁿ(k)), TS(S₁ ^out(k))) of the node (MA, SL1, SL2) sending the synchronization message (SY(k));

    - for each slave node (SL1, SL2) the master counter value (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k)) for a measured slave counter, value (TS(S₁ ⁱⁿ(k)), TS(S₂ ⁱⁿ(k)) is estimated by using a state estimation method applied to a probabilistic model for state variables of the nodes (MA, SL1, SL2), wherein each slave clock is synchronized based on the estimated master counter value (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k)), wherein the probabilistic model includes

    - a state transition model describing the dependency of state variables (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k), R₁(k), R₂(k)) of slave nodes (SL1, SL2) in a synchronization cycle on state variables of slave nodes (SL1, SL2) in the previous synchronization cycle;

    - an observation model describing the dependency of known variables (TS(M(k)), b_n(k), C_n(k)) of master and slave nodes (MA, SL1, SL2) on state variables (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k), R₁(k), R₂(k)) of slave nodes (SL1, SL2) ;

    and wherein the state variables (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k), R₁(k) ,
    R₂(k)) of the slave nodes (MA, SL1, SL2) are estimated by the estimation method,
    characterized in that the estimation method is based on a Kalman-Filter, in which the estimation of the state variables is based on variables of all nodes, wherein in each synchronization cycle the known variables (TS(M(k)), b_n(k), C_n(k)) of each node (MA, SL1, SL2) are transmitted to a common fusion center (FC) in which based on the state transition model and the observation model the state variables (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k), R₁(k), R₂(k)) of each slave node (MA, SL1, SL2) are estimated by the estimation method, said estimated state variables (x₁ ⁱⁿ(k), x₂ ⁱⁿ(k), R₁(k), R₂(k)) being transmitted to the corresponding nodes (MA, SL1, SL2) in order to synchronize the slave clocks.
19. Communication network according to claim 18, which is adapted to perform a method according to one of claims 2 to 17.

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